Classification of certain pairs of operators (P, Q) satisfying [P, Q] = −i Id

In a separable Hilbert space a certain class of pairs of operators (P, Q) satisfying the Born-Heisenberg commutation relation [P, Q] = ?i Id on a dense domain Ω is investigated. This class is essentially defined by requiring Q bounded and self-adjoint, P symmetric and $\text{PΩ ? Ω, QΩ ? Ω}$. We show that Q is absolutely continuous and that P can be thought of as a first order differential operator. The class considered contains the pair “angle ?” and “angular momentum L_z.” It is expected that the methods of this paper can be applied to more general classes of operators (P, Q) including the Schrödinger case.     

On the period function of x″+f(x)x′+g(x)=0

We study the period function T of a center O of the title's equation. A sufficient condition for the monotonicity of T, or for the isochronicity of O, is given. Such a condition is also necessary, when f and g are odd and analytic. In this case a characterization of isochronous centers is given. Some classes of plane systems equivalent to such equation are considered, including some Kukles’ systems.     

Common fixed points for (g, f) type maps in cone metric spaces

In this paper we study common fixed points for the self and non-self (g, f) type maps in cone metric spaces. Our main results are related to the cases when g is f quasi contraction in a sense of Das and Naik, and the cone need not be normal. These results generalize several well known comparable results in the literature.     

The differential equation x = f ⊥ x

A theory is presented for absolutely continuous solutions of the general scalar first order autonomous o.d.e. Necessary and sufficient conditions are given for local and global existence and uniqueness, and further topics include existence-uniqueness duality, structure of the solution set and weak solutions.     

Factorization of Q(h(T)(x)) over a finite field where Q(x) is irreducible and h(T)(x) is linear—I

Let GF(q) be the finite field of order q, let Q(x) be an irreducible polynomial in GF(q)(x), and let h(T)(x) be a linear polynomial in GF(q)[x], where T:x→x^q. We use properties of the linear operator h(T) to give conditions for Q(h(T)(x)) to have a root of arbitrary degree k over GF(q), and we describe how to count the irreducible factors of Q(h(T)(x)) of degree k over GF(q). In addition we compare our results with those Ore which count the number of irreducible factors belonging to a linear polynomial having index k.     

P-points and Q-points over a measurable cardinal

This paper presents constructions of κ-ultrafilters over a measurable cardinal κ, specifically p-points and q-points, in the continuation of works of Gitik, Kanamori and Ketonen. On the assumption of supercompactness, Kunen has shown the existence of a chain of p-points, in the Rudin–Keisler ordering, of maximal length. We shall prove a similar result for q-points. Results of Mitchell [8,9] establish connections between Rudin—Keisler chains of κ-ultrafilters and inner models of “?ν(o(ν)= ν⁺⁺)”. This shows the necessity of some strong large cardinal hypothesis. The second part of the paper is devoted to a separation property of κ- ultrafilters (cf. Kanamori and Taylor). To answer a question of Taylor concerning the existence of a non-separating p-point, we use a combination of Silver's Forcing and iterated ultrapowers; the proof itself may be of some interest.     

Howell designs of type H(P − 1, P + 1)

If p > 3 is a prime, it is shown that a Howell design of type H(p ? 1, p + 1) can be constructed. When p = 5 or p > 3 and p ≡ ±3, ±13 (mod 40), the construction can be accomplished by an extension of the familiar strong starter method used to build Room squares.     

On the solutions to x′(t) = A(t) x (t) over all A(t), where P ⩽ A(t) ⩽ Q

Let P_ij and q_ij be positive numbers for i ≠ j, i, j = 1, …, n, and consider the set of matrix differential equations x′(t) = A(t) x(t) over all A(t), where a_ij(t) is piecewise continuous, a_ij(t) = ?∑_{i ≠ j}a_ij(t), and p_ij ? a_ij(t) ? q_ij all t. A solution x is also to satisfy ∑_{i = 1}ⁿx_i(0) = 1. Let C_t denote the set of all solutions, evaluated at t to equations described above. It is shown that $\text{C}{}_{\mathrm{t}}$, the topological closure of C_t, is a compact convex set for each t. Further, the set valued function $\text{C}{}_{\mathrm{t}}$, of t is continuous and $\text{limit}{}_{\mathrm{t\; \to \; \infty }}\text{C}\text{?}{}_{\mathrm{t}}\text{= ∩}\text{C}\text{?}{}_{\mathrm{t}}$.     

Inequalities between ∥f(A + B)∥ and ∥f(A) + f(B)∥

The conjecture posed by Aujla and Silva [J.S. Aujla, F.C. Silva, Weak majorization inequalities and convex functions, Linear Algebra Appl. 369 (2003) 217-233] is proved. It is shown that for any m-tuple of positive-semidefinite n × n complex matrices A_j and for any non-negative convex function f on [0, ∞) with f(0) = 0 the inequality ?f(A₁) + f(A₂) + ? + f(A_m)? ? ? f(A₁ + A₂ + ? + A_m)? holds for any unitarily invariant norm ? · ?. It is also proved that ?f(A₁) + f(A₂) + ? + f(A_m)? ? f(?A₁ + A₂ + ? + A_m?), where f is a non-negative concave function on [0, ∞) and ? · ? is normalized.     

On the diophantine equation x(x + 1)(x + 2)…(x + (m − 1)) =g(y)

Let g(y) ? Q[Y] be an irreducible polynomial of degree n ≥ 3. We prove that there are only finitely many rational numbers x, y with bounded denominator and an integer m ≥ 3 satisfying the equation x(x + 1) (x + 2)…(x + (m − 1) ) = g(y). We also obtain certain finiteness results when g(y) is not an irreducible polynomial.     

Factorization of Q(h(T)(x)) over a finite field,where Q(x) is irreducible and h(T)(x) is linear II

Let GF(q) be the finite field of order q, let Q(x) be an irreducible polynomial in GF(qⁱ)[x], and let H(x) = h(T)(x) be a linear polynomial in GF(q)[x]. We give the degrees of the irreducible factors of Q(H(x)) in GF(qⁱ)[x], and the number of irreducible factors of each degree. We consider the special cases when H(x) is a trace function, and when h(x) is cyclotomic. Finally, we give several examples.